We now multiply by \(\prod _{2<p\le y}(1-2/p)\) , apply (REF ) and observe \(\Psi (H/2,y) \sim \Psi (H,y)/2\)  by [1]}.
In the case \(\log \log H/\log y \sim a<1/2\) , (REF ) still holds. Moreover, \(1-\alpha = \log \log H/\log y+o(1) = a + o(1)\) . As \(\Psi (H/2,y) \sim \Psi (H,y)2^{-\alpha +o(1)}\)  by [1]} we see that
\( \sum _{r \ge 1} g_y(r)\left\lbrace  \frac{H}{2r}\right\rbrace  \left(1-\left\lbrace  \frac{H}{2r}\right\rbrace \right)\sim \frac{-2\log y}{\log \log H} H_y(\alpha )\zeta (-a) \Psi (H,y) 2^{-\alpha } .\) 
